He said: “These people poison the game, their comments are quite abhorrent, they can unsettle players, there’s no doubt about it.

“And as we have seen in some of the political incidents and the tragic death, trolling can end up with horrendous circumstances.

“We do know if somebody is doing it on YouTube or Facebook one day they could be in your back garden the next day with a particular weapon.

“I don’t think we are taking it seriously enough.

“The other thing is to appeal to people and say ‘this is a fantastic game, it’s the centre of our culture, we love the game with a passion and we are not going to stand a small number of psychopaths and fellow travellers who don’t deserve any attention’.

“We should be able to give the police and the security services and other people the authority to actually tackle it hard.

“I would make an appeal to some people who are not in that category but just a bit crazed because of disappointment to look at the pitch and look at your team. What happens off the pitch is all about you and if you want to be a good fan you don’t do this.”

McLeish conducted a root and branch review of the Scottish game in 2010 and now sits on the board of the Scottish Football Supporters Association.

Speaking to BBC Radio Scotland, he backed plans by the Conservative government at Westminster to legally force social media giants like Facebook and Twitter to report to the police any threats of physical harm and to delete any posts that could incite violence.

He said: “We have got to stamp it out and it’s a bigger issue than just football.

“I don’t often support Conservative governments but they are quite keen to get involved with some of the big tech companies and basically say if they don’t improve what they are doing then we will legislate. I think the Tories at Westminster are beginning to take it seriously.

“In newspapers and programmes they are things you can say and things you can’t say so why can’t that be applied to the internet.”

After the tweet was sent to Brown last week, Burns’s son Jonathan told followers on Twitter to “pray” for the sick troll.

He tweeted: “It’s a pity people have to sink to this level but without the proper upbringing and basic human qualities, this is what you get. Pray for them.”

Brown’s Celtic team mate Scott Sinclair has been targeted by racist abuse online several times and police launched a fresh probe into a tweet targeted at him last month.